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3   SPECTRUM OF A COMMUTATIVE BANACH ALGEBRA       331

with exponents /c^O. To each function fe <3^p(^) there corresponds the analytic
function in the disk | z | < 1 defined by

F(z)

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For 1 ^p  < oo (resp. p = -f oo) /is the limit (resp. weak limit) in ^"(ju-) of the
functions 0t->F(reie) as n->l, so that Hp(/x) may be identified with the space of
analytic functions F corresponding to the functions /e Jfp(/x). (Use the fact that
^P(JJL) is the closure (resp. weak closure) of B in ^(ju,).) In particular, H2(^) is
identified with the space of functions

which are analytic for \z\<\ and such that JT \cn\2 < +00 (cf. the problem in

n = 0

Section 9.13).

(b)    Let v be a measure disjoint from /x which is quasi-representative for %. Show
that e~t$ - v is also a quasi-representative measure for #, and hence by induction that
e-nio . v is a quasi-representative measure for %, for all integers n > 0. Hence show that
v = 0 and consequently (Problem 14) that every quasi-representative measure for x
has base ju,.

(c)    Let/e 3^l(\L} be nonnegligible (with respect to fx). Then the function log|/|
is ju,-integrable, or equivalently, JM(|/|) > 0, and consequently / is the product of an
exterior function and an interior function. (This follows from Problem ll(b) if

\fdfji 7* 0. If \fdfji = 0 then/, identified with an analytic function on the unit disk,

is of the form /(z) = zng(z) where n > 0 and 0(0) ^ 0. We have g e ^'(ju.) and
log |/| = log \g\ on U.) In particular, a function belonging to ^(p) which vanishes
on a non-ju,-negligible subset of U is /z-negligible. Every function in ^(/u.) is the
product of two functions in .^2(/x) and conversely (Problem 15(e)). A function
/^> 0 belonging to >l(p) is equivalent to a function of the form \g \2, where g e Jf 2(ju,),
if and only if JM(/) > 0 (or equivalently if and only if log |/| is p-integrable). (See
Section 22.19, Problem 19.)

17.    Let B be the Banach algebra considered in Problem 16, and BI the Banach subalgebra
of B consisting of functions/continuous on |z| <j 1 and analytic on |z| < 1, and such
that/'(0) = 0. For each complex number c such that c\ <j 1, let ftc denote the measure
Fc  /x, where Fc(z) = 1  ^(cz). Show that the measures /xc are representative meas-
ures for the character ^ : /t/(0); the representative measures which are extremal
points of 9R(x) are *ne Me suc^ that |c|  1; the measure p,o = p, is the unique Jensen
measure for xi and hence there exists no hyperextremal representative measure for x*

18.    Let A be a commutative Banach algebra with unit element.

(a)    Let u e A be such that p(u) < 1. Then for each character ^ of A the element
1  x(w)w is invertible. If v  (x(u)  *0(1  x(") w)-1 tnen 1x^)1 ^ ^>

(b)    If xi, X* are two characters of A, let CT(XI, Xa) denote the least upper bound of
the number IxaWI as u runs through the set of elements of A such that p(u) <J 1
and xi(w) = 0. Show that if p(u)  1 we havethe closure of B0M in ^(p), deduce that/i